# What's coherent about coherent sheaves?

In a recent answer to a recent question, BCnrd wrote

[...] beyond the coherent case one cannot expect information about a fiber (e.g., vanishing, 6 generators, etc.) to "propogate" to information in a neighborhoood (which would be the spirit behind the choice of word "coherent", I suppose). [...]

Is that what motivates the adjective coherent? Is this documented somewhere?

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As far as I can tell, the earliest reference is Seminaire Ecole Normale et Superieur 1951-52 exposes XVIII et XIX. This is given in H. Cartan, Serre Un théorème de finitude concernant les variétés analytiques compactes C. R. Acad. Sci. Paris 237, (1953). 128–130. All of this work seems to be in analytic spaces, so perhaps there is something in that area that will be a clue. – David Roberts Oct 29 '10 at 4:37
My recollection is that in the (awesome) book "Coherent analytic sheaves", the historical comments either in the Introduction or the appendix on "yoga of coherence" refer to coherence as a kind of principle of local analytic continuation. Take a look at those parts of that book and you may find such documentation that you seek. Maybe also look at historical notes in Oka's collected works. – BCnrd Oct 29 '10 at 7:00

## 1 Answer

Looking at the paper of MALATIAN

"Faisceaux analytiques: étude du faisceau des rélations entre p fonctions holomorphes",

Séminaire Henry Cartan, tome 4 (1951-52), exp. n.15, p. 1-10

one finds the

Definition 3

"On dit qu'un sous-faisceau analytique $\mathcal{F}$ de $\mathcal{O}_E^q$ is $cohérent$ au point $x \in E$, s'il existe un voisinage ouvert $U$ de $x$ et un système fini d'elements $u_i \in \mathcal{O}_U^q$ jouissant de la propriété suivante: pour tout $y \in U$, le sous-module de $\mathcal{O}_U^q$ engendré par les $u_i$ est précisement $\mathcal{F}_y$. On dit qu'un faisceau $\mathcal{F}$ est cohérent (tout court) s'il est coherent en tout point de $E$."

And, in the following page:

"...En d'autre termes, cette condition exprime que le faisceau $induit$ par $\mathcal{F}$ sur l'ouvert $U$ est "engendré" par un sous-module de $\mathcal{O}_U^q$."

Reading this, it seems that the original definition given by Cartan in its seminar is somehow related to the "coherent behaviour" of $\mathcal{F}$ as a subsheaf of $\mathcal{O}_U^q$, in terms of generation of the stalks.

EDIT.

However, this is not the whole story. Loking at the introduction of the book of Grauert-Remmert, as Brian suggests, it appears that the word "coherent" was actually introduced by Cartan some years before, in the middle of the '40; in fact, he investigated the so-called "coherent systems of punctual modules" when studyng the Cousin's problem. But he does not mention this previous work in his Seminar, when he introduces coherent analytic sheaves.

Grauert-Remmert write that

"coherence is, in a vague sense, a local principle of analytic continuation".

And Cartan himself, in its collected works, says

"En gros on peut dire que, pour en $A$-faisceaux $\mathcal{F}$ cohérent en un point $a$ de $A$, la connaissance du module $\mathcal{F}_a$ détermine les modules $\mathcal{F}_x$ attachées aux points $x$ suffisamment voisins de $a$."
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Thanks! $\mbox{}$ – Mariano Suárez-Alvarez Oct 29 '10 at 18:35